Theorem prnz 4741

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4726	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4293	1 ⊢ {𝐴, 𝐵} ≠ ∅

This proof depends on syntax axioms:   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453  ∅c0 4282  {cpr 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-un 3907  df-nul 4283  df-sn 4588  df-pr 4590

This theorem is used by:  opnz  5453  propssopi  5489  fiint  9300  wilthlem2  27313  upgrbi  29558  wlkvtxiedg  30092  shincli  31851  chincli  31949  constrextdg2lem  34266  spr0nelg  48384  sprvalpwn0  48391